An Empirical Model Predicting the Viscosity of Highly Concentrated
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Rheol Acta &2001) 40: 434±440 Ó Springer-Verlag 2001 ORIGINAL CONTRIBUTION Burkhardt Dames An empirical model predicting the viscosity Bradley Ron Morrison Norbert Willenbacher of highly concentrated, bimodal dispersions with colloidal interactions Abstract The relationship between the particles. Starting from a limiting Received: 7 June 2000 Accepted: 12 February 2001 particle size distribution and viscos- value of 2 for non-interacting either ity of concentrated dispersions is of colloidal or non-colloidal particles, great industrial importance, since it e generally increases strongly with is the key to get high solids disper- decreasing particle size. For a given sions or suspensions. The problem is particle system it then can be treated here experimentally as well expressed as a function of the num- as theoretically for the special case ber average particle diameter. As a of strongly interacting colloidal consequence, the viscosity of bimo- particles. An empirical model based dal dispersions varies not only with on a generalized Quemada equation the size ratio of large to small ~ / Àe g g 1 À / is used to describe particles, but also depends on the g as a functionmax of volume fraction absolute particle size going through for mono- as well as multimodal a minimum as the size ratio increas- dispersions. The pre-factor g~ ac- es. Furthermore, the well-known counts for the shear rate dependence viscosity minimum for bimodal dis- of g and does not aect the shape of persions with volumetric mixing the g vs / curves. It is shown here ratios of around 30/70 of small to for the ®rst time that colloidal large particles is shown to vanish if interactions do not show up in the colloidal interactions contribute maximum packing parameter and signi®cantly. /max can be calculated from the B. Dames á B. R. Morrison particle size distribution without N. Willenbacher &&) further knowledge of the interac- Key words Bimodal dispersions á BASF Aktiengesellschaft tions among the suspended particles. Rheology á Generalized Quemada Polymers Laboratory 67056 Ludwigshafen, Germany On the other hand, the exponent e is equation á Maximum packing e-mail: [email protected] controlled by the interactions among fraction á Colloidal interactions Introduction reasonable stir-, pump-, and sprayability during trans- port and application. Particle loadings beyond a volume Market competition, cost reduction, and environmental fraction of 0.60 can only be achieved if the particle size driving forces are pushing industrial researchers to distribution is either broad or bi- or multimodal. formulate polymer dispersions with a particle loading Consequently, the rheology and the packing possibilities as high as possible. The bene®t will be an increased time- of highly concentrated suspensions have been discussed space yield during production, reduced transportation intensively by academic as well as industrial researchers. costs, and less energy consumption for the removal of The literature strongly focuses on suspensions of non- solvent in the application process. At the same time the Brownian spheres &e.g., Farris 1968; Chong et al. 1971; viscosity has to be kept low enough in order to ensure a Poslinski et al. 1988; Sengun and Probstein 1989, 1997; sucient heat transfer during polymerization as well as Gondret and Petit 1997) or on Brownian but so-called 435 hard spheres &Rodriguez and Kaler 1992; Chang and size, and that the viscosity minimum for bimodal Powell 1994; Shikata et al. 1998). Only a few papers deal dispersions with volumetric mixing ratios of around with the rheology of Brownian particle suspensions with 30/70 of small to large particles vanishes, if colloidal either electrostatic &Zaman and Moudgil 1999; Richter- interactions contribute signi®cantly. ing and MuÈ ller 1995) or steric interactions &Woutersen and de Kruif 1993; D'Haene and Mewis 1994; Green- wood et al. 1997) or a combination of both &Homan Experimental 1992; Chu et al. 1998). Monodisperse suspensions can in principle pack in Samples several ways with dierent maximum packing fractions, All dispersions were made by seeded emulsion polymerization. For e.g., /m 0.52 for simple cubic, /m 0.74 for hexagonal our examinations, we used dispersions with a monomodal as well or 0.63 for random close packing. For a random close as dispersions with a bimodal particle size distribution. In order to packed bimodal system a maximum packing fraction produce bimodal dispersions with high solids content we started the polymerization with two dierent seeds diering in their average /m 0.87 is achieved assuming Rlarge/Rsmall >10so that small particles together with the solvent molecules particle size. The initial amount of these seeds was governed by the desired particle size distribution. This is a convenient method to can be treated as a continuum. The small particle obtain bimodal dispersions with a very high viscosity because there volume fraction then is ns 0.27 &Farris 1968). From is no need for mixing two monomodal dispersions and evaporation simple geometrical considerations it follows that the of water afterwards that could induce coagulation and agglomer- small particles exactly ®t into the pores between the large ation. All dispersions had the same chemical composition consist- ing of 99 parts butylacrylate, 1 part acrylic acid, 0.8 parts ones at a critical size ratio Rlarge/Rsmall 6.46 &McGeary dodecylphenoxybenzene disulfonic acid sodium salt &Dowfax 1961). In this case the number ratio of small to large 2A1), and 0.2 parts sodium laurylsulfate. particles Nsmall/Nlarge 100; that means there are much The initiation was done either with sodiumperoxodisulfate, a so- more small particles than interstices between large called thermal initiation, or with the redox system ascorbic acid/ tert-butylhydroperoxide. Thus we attained dispersions with the particles. Therefore, the simple picture of small particles same particle size distribution but with dierent surface properties. ®lling the space between the large ones, thus providing a high maximum packing fraction and a low viscosity, is Measurements not valid for bimodal suspensions with size ratios like this. If the size ratio is below this critical value the lattice Viscosity measurements were done with a controlled strain of the large particles may be expanded, and microphase rotational rheometer Rheometrics RFS II equipped with a Couette separation or the formation of superstructures can also &Ri 16 mm, Ra 17 mm) sample cell. All measurements were performed at 20 °C. occur. These phenomena have been discussed intensively Particle size distributions were determined using the analytical in the literature &e.g., Verhaegh and Lekkerkerker 1997; ultracentrifuge technique &MaÈ chtle 1992). Typical results are Bartlett and Pusey 1993). Despite the variety of packing presented in Fig. 1, showing the dierential mass distribution as possibilities, viscosity reduction in bimodal suspensions a function of particle diameter. The mean size of each particle species was determined from the position of the maximum of the is a very general phenomenon and has been observed for corresponding peak, and the relative fraction of each species was large as well as intermediate and small size ratios. A calculated from the area under each peak. minimum viscosity at a small particle volume fraction ns of about 0.3 has been reported for suspensions of non- Brownian glass beads &Chong et al. 1971) as well as for Results and discussion Brownian hard sphere &Rodriguez and Kaler 1992) and colloidally interacting suspensions &Zaman and Moudgil Experiments 1999). The aim of this paper is to work out a procedure The viscosity of a large number of mono- and bimodal which allows for a simple calculation of the viscosity of dispersions was determined as a function of shear rate suspensions or dispersions at particle loadings and shear and volume fraction. Data for a monomodal system rates relevant for industrial applications. An empirical with 250 nm particle diameter are shown in Fig. 2. model based on a generalized Quemada equation is used These results are typical for all suspensions investigated to describe g as a function of volume fraction for mono- here. Varying the particle concentration, two regimes as well as multimodal dispersions. This model quanti- diering with respect to the shape of the viscosity curves tatively describes the viscosity of a large number of can be distinguished. bimodal suspensions with signi®cantly dierent size and At volume fractions / below a critical packing mixing ratios investigated here. Moreover, it qualita- fraction /c the viscosity curves exhibit a Newtonian tively predicts well-known features of suspensions with plateau at low shear rates followed by a shear thinning colloidal interactions. In particular, it is shown, that the region, indicating that the dispersions behave like a viscosity varies not only with the size ratio of large to liquid. The onset of shear thinning is shifted to higher small particles, but also depends on the absolute particle shear rates and the degree of shear thinning decreases 436 Fig. 1 Dierential particle size distribution from AUC mea- surements for dierent bimodal dispersions. The labels inserted in the diagrams give Dlarge and Dsmall in nm and the small particle volume fraction times 100 &i.e., B-915/310±10 means Dlarge 915 nm, Dsmall 310 nm, and ns 0.10) Fig. 2 Viscosity vs shear rate for a monomodal dispersion &diameter: 250 nm) at dierent particle loadings with decreasing volume fraction. The high shear limiting curves. Therefore, the relative change of viscosity with value of the viscosity is not reached within the shear rate volume fraction is independent of shear rate and it is range &c_ < 500 sÀ1) investigated here. sucient to discuss the eect of packing density on For / > /c the dispersions are shear thinning in the viscosity at a ®xed shear rate. whole shear rate range investigated and neither a high Dispersions go through various ¯ow ®elds during nor a low shear plateau is observed.